Tangent vector to a point on the surface. Recently, I studied the intrinsic geometry of surfaces. Is tangent vector to a point on the surface something intrinsic? (In another word, is the tangent plane independent of the ambient space?) I guess it is intrinsic but I only need clarification. 
 A: Tangent vectors to a point on a manifold (or in particular a surface) are indeed intrinsic. An abstract manifold $M$ is a topological space which is Hausdorff, second countable, and locally homeomorphic to Euclidean space. If we want our manifold to be smooth, we have to specify which functions are smooth. We call the global smooth functions $\mathscr{C}^\infty(M)$.
Supposing we have these data, we can define abstractly the tangent space at a point $p\in M$ as follows. An operator $D:\mathscr{C}^\infty(M)\to \mathbb{R}$ is called a derivation of $\mathscr{C}^\infty(M)$ at $p$ if it is $\mathbb{R}-$linear and satisfies the following Leibniz rule: given $f,g\in \mathscr{C}^\infty(M)$, 
$$ D(fg)=D(f)g(x)+f(x)D(g).$$
The space of such derivations of $\mathscr{C}^\infty(M)$ at $p$ is called the tangent space of $M$ at $p$, and denoted $T_pM$.
Here is an example: if $M=\mathbb{R}^n$ (or any open subset of $\mathbb{R}^n)$, then an example of a derivation at $p\in M$ is the partial derivative operator $\frac{\partial}{\partial x^i}|_p$. It turns out that a basis of $T_pM$ as described above is $(\frac{\partial}{\partial x^1}|_p,\ldots, \frac{\partial}{\partial x^n}|_p)$. This is canonically isomorphic to the naïve tangent space to $M$ at $p$ given by the "tangent directions." Indeed, one can see $\frac{\partial}{\partial x^i}$ as being the tangent direction pointing in the $x^i$ coordinate direction.
One can show in general that so-defined $T_pM$ is a vector space such that $\dim T_pM=\dim M$. Moreover, it has a basis consisting of partial derivative operators defined using the coordinate charts on the manifold. This is a valid and embedding-independent notion of tangent space for a manifold.
You can find on wikipedia another notion defined in terms of trajectories of curves.
